Gregersen E. (editor) The Britannica Guide to Statistics and Probability
Подождите немного. Документ загружается.
7 The Britannica Guide to Statistics and Probability 7
250
degree (1721). In 1723–24 he wrote Exercitationes quaedam
Mathematicae on differential equations and the physics of
flowing water, which won him a position at the influential
Academy of Sciences in St. Petersburg, Russia. Bernoulli
lectured there until 1732 in medicine, mechanics, and
physics, and he researched the properties of vibrating and
rotating bodies and contributed to probability theory. In
that same year he returned to the University of Basel to
accept the post in anatomy and botany. By then he was
widely esteemed by scholars and also admired by the pub-
lic throughout Europe.
Daniel’s reputation was established in 1738 with
Hydrodynamica, in which he considered the properties
of basic importance in fluid flow, particularly pressure,
density, and velocity, and set forth their fundamental
relationship. He put forward what is called Bernoulli’s
principle, which states that the pressure in a fluid
decreases as its velocity increases. He also established
the basis for the kinetic theory of gases and heat by
demonstrating that the impact of molecules on a surface
would explain pressure and that, assuming the constant,
random motion of molecules, pressure and motion
increase with temperature. About 1738 his father pub-
lished Hydraulica. An aparent attempt to obtain priority
for himself, this was yet another instance of his antago-
nism toward his son.
Between 1725 and 1749 Daniel won 10 prizes from the
Paris Academy of Sciences for work on astronomy, gravity,
tides, magnetism, ocean currents, and the behaviour of
ships at sea. He also made substantial contributions in
probability. He shared the 1735 prize for work on plane-
tary orbits with his father, who, it is said, threw him out of
the house for thus obtaining a prize he felt should be his
alone. Daniel’s prizewinning papers reflected his success
on the research frontiers of science and his ability to set
251
7
Biographies 7
forth clearly before an interested public the scientific
problems of the day. In 1732 he accepted a post in botany
and anatomy at Basel; in 1743, one in physiology; and in
1750, one in physics.
Jakob beRnoulli
(b. Jan. 6, 1655 [Dec. 27, 1654, Old Style], Basel, Switz.—d. Aug. 16,
1705, Basel)
The first of the Bernoulli family of Swiss mathematicians
was Jakob Bernoulli. He introduced the first principles of
the calculus of variation. Bernoulli numbers, a concept
that he developed, were named for him.
The scion of a family of drug merchants, Jakob
Bernoulli was compelled to study theology but became
interested in mathematics despite his father’s opposition.
His travels led to a wide correspondence with mathemati-
cians. Refusing a church appointment, he accepted a
professorial chair of mathematics at the University of
Basel in 1687. Following his mastery of the mathematical
works of John Wallis, Isaac Barrow (both English), René
Descartes (French), and G.W. Leibniz, who first drew his
attention to calculus, he embarked on original contribu-
tions. In 1690 Bernoulli became the first to use the term
“integral” in analyzing a curve of descent. His 1691 study
of the catenary, or the curve formed by a chain suspended
between its two extremities, was soon applied in building
suspension bridges. In 1695 he also applied calculus to the
design of bridges. During these years, he often engaged in
disputes with his brother Johann Bernoulli over mathe-
matical issues.
Jakob Bernoulli’s pioneering work Ars Conjectandi
(published posthumously, 1713; “The Art of Conjecturing”)
contained many of his finest concepts: his theory of per-
mutations and combinations; the so-called Bernoulli
7 The Britannica Guide to Statistics and Probability 7
252
Among other things, Jakob (known also as Jacques) Bernoulli is renowned
for the Bernoulli numbers and the Bernoulli law of large numbers. SSPL via
Getty Images
253
7
Biographies 7
numbers, by which he derived the exponential series; his
treatment of mathematical and moral predictability; and
the subject of probability—containing what is now called
the Bernoulli law of large numbers, basic to all modern
sampling theory. His works were published as Opera Jacobi
Bernoullii, 2 vol. (1744).
bha¯ skaRa ii
(b. 1114, Biddur, India—d. c. 1185, probably Ujjain)
Bhāskara II was the leading mathematician of the 12th
century and wrote the first work with full and systematic
use of the decimal number system.
Bhāskara II was the lineal successor of the noted
Indian mathematician Brahmagupta (598–c. 665) as head
of an astronomical observatory at Ujjain, the leading
mathematical centre of ancient India.
In his mathematical works, particularly Līlāvatī (“The
Beautiful”) and Bījagan
·
ita (“Seed Counting”), he not only
used the decimal system but also compiled problems from
Brahmagupta and others. He filled many gaps in
Brahmagupta’s work, especially in obtaining a general
solution to the Pell equation (x
2
= 1 + py
2
) and in giving
many particular solutions. Bhāskara II anticipated the
modern convention of signs (minus by minus makes plus,
minus by plus makes minus) and evidently was the first to
gain some understanding of the meaning of division by
zero. He specifically stated that the value of
3
/
0
is an infi-
nite quantity, but his understanding seems to have been
limited, for he also wrongly stated that
a
⁄
0
× 0 = a. Bhāskara
II used letters to represent unknown quantities, much as
in modern algebra, and solved indeterminate equations of
1st and 2nd degrees. He reduced quadratic equations to a
single type and solved them and investigated regular
7 The Britannica Guide to Statistics and Probability 7
254
polygons up to those having 384 sides, thus obtaining a
good approximate value of π = 3.141666.
In other of his works, notably Siddhāntaśiroman
·
i
(“Head Jewel of Accuracy”) and Karan
·
akutūhala (“Calculation
of Astronomical Wonders”), he wrote on his astronomi-
cal observations of planetary positions, conjunctions,
eclipses, cosmography, geography, and the mathematical
techniques and astronomical equipment used in these
studies. Bhāskara II was also a noted astrologer, and tra-
dition has it that he named his first work, Līlāvatī, after
his daughter to console her when his astrological med-
dling, coupled with an unfortunate twist of fate, is said
to have deprived her of her only chance for marriage and
happiness.
ludwig eduaRd bolTzMann
(b. Feb. 20, 1844, Vienna, Austria—d. Sept. 5, 1906, Duino, Italy)
The physicist Ludwig Eduard Boltzmann had his greatest
achievement in the development of statistical mechanics,
which explains and predicts how the properties of atoms
(such as mass, charge, and structure) determine the visible
properties of matter (such as viscosity, thermal conductiv-
ity, and diffusion).
After receiving his doctorate from the University of
Vienna in 1866, Boltzmann held professorships in mathe-
matics and physics at Vienna, Graz, Munich, and Leipzig.
In the 1870s Boltzmann published a series of papers in
which he showed that the second law of thermodynam-
ics, which concerns energy exchange, could be explained
by applying the laws of mechanics and the theory of prob-
ability to the motions of the atoms. In so doing, he made
clear that the second law is essentially statistical and that
a system approaches a state of thermodynamic equilib-
rium (uniform energy distribution throughout) because
255
7
Biographies 7
equilibrium is overwhelmingly the most probable state of
a material system. During these investigations Boltzmann
worked out the general law for the distribution of energy
among the various parts of a system at a specific tempera-
ture and derived the theorem of equipartition of energy
(Maxwell-Boltzmann distribution law). This law states
that the average amount of energy involved in each differ-
ent direction of motion of an atom is the same. He derived
an equation for the change of the distribution of energy
among atoms resulting from atomic collisions and laid the
foundations of statistical mechanics.
Boltzmann was also one of the first continental scien-
tists to recognize the importance of the electromagnetic
theory proposed by James Clerk Maxwell of England.
Though his work on statistical mechanics was strongly
attacked and long misunderstood, his conclusions were
finally supported by the discoveries in atomic physics that
began shortly before 1900 and by recognition that fluctu-
ation phenomena, such as Brownian motion (random
movement of microscopic particles suspended in a fluid),
could be explained only by statistical mechanics.
geoRge boole
(b. Nov. 2, 1815, Lincoln, Lincolnshire, Eng.—d. Dec. 8, 1864,
Ballintemple, County Cork, Ire.)
The English mathematician George Boole helped estab-
lish modern symbolic logic and whose algebra of logic,
now called Boolean algebra, is basic to the design of digital
computer circuits.
Boole was given his first lessons in mathematics by his
father, a tradesman, who also taught him to make optical
instruments. Aside from his father’s help and a few years
at local schools, however, Boole was self-taught in mathe-
matics. When his father’s business declined, George had
7 The Britannica Guide to Statistics and Probability 7
256
to work to support the family. From the age of 16 he taught
in village schools in the West Riding of Yorkshire, opening
his own school in Lincoln when he was 20. During scant
leisure time he read mathematics journals in the Lincoln’s
Mechanics Institute. There he also read Isaac Newton’s
Principia, Pierre-Simon Laplace’s Traité de mécanique céleste,
and Joseph-Louis Lagrange’s Mécanique analytique and
began to solve advanced algebra problems.
Boole submitted a stream of original papers to the new
Cambridge Mathematical Journal, beginning in 1839 with his
“Researches on the Theory of Analytical Transformations.”
These papers were on differential equations and the alge-
braic problem of linear transformation, emphasizing the
concept of invariance. In 1844, in an important paper in
the Philosophical Transactions of the Royal Society for which
he was awarded the Royal Society’s first gold medal for
mathematics, he discussed how methods of algebra and
calculus might be combined. Boole soon saw that his alge-
bra could also be applied in logic.
Developing novel ideas on logical method and con-
fident in the symbolic reasoning he had derived from
his mathematical investigations, he published in 1847 a
pamphlet, “Mathematical Analysis of Logic,” in which
he argued persuasively that logic should be allied with
mathematics, not philosophy. He won the admiration
of the English logician Augustus De Morgan, who pub-
lished Formal Logic the same year. On the basis of his
publications, Boole in 1849 was appointed professor of
mathematics at Queen’s College, County Cork, even
though he lacked a university degree. In 1854 he published
An Investigation into the Laws of Thought, on Which Are
Founded the Mathematical Theories of Logic and Probabilities,
which he regarded as a mature statement of his ideas. The
next year he married Mary Everest, niece of Sir George
257
7
Biographies 7
Everest, for whom the mountain is named. The Booles
had five daughters.
One of the first Englishmen to write about logic, Boole
pointed out the analogy between algebraic symbols and
those that can represent logical forms and syllogisms,
showing how the symbols of quantity can be separated
from those of operation. With Boole in 1847 and 1854
began the algebra of logic, or what is now called Boolean
algebra. Boole’s original and remarkable general symbolic
method of logical inference, fully stated in Laws of Thought
(1854), enables one, given any propositions involving any
number of terms, to draw conclusions that are logically
contained in the premises. He also attempted a general
method in probabilities, which would make it possible
from the given probabilities of any system of events to
determine the consequent probability of any other event
logically connected with the given events.
In 1857 Boole was elected a fellow of the Royal Society.
The influential Treatise on Differential Equations appeared
in 1859 and was followed the next year by its sequel, Treatise
on the Calculus of Finite Differences. Used as textbooks for
many years, these works embody an elaboration of Boole’s
more important discoveries. Boole’s abstruse reasoning
has led to applications of which he never dreamed. For
example, telephone switching and electronic computers
use binary digits and logical elements that rely on Boolean
logic for their design and operation.
giRolaMo caRdano
(b. Sept. 24, 1501, Pavia, duchy of Milan [Italy]—d. Sept. 21,
1576, Rome)
The Italian physician, mathematician, and astrologer
Girolamo Cardano wrote a book, Ars magna (The Great
7 The Britannica Guide to Statistics and Probability 7
258
Art; or, The Rules of Algebra), that is one of the cornerstones
in the history of algebra.
Educated at the universities of Pavia and Padua,
Cardano received his medical degree in 1526. In 1534 he
moved to Milan, where he lived in great poverty until he
became a lecturer in mathematics. Admitted to the col-
lege of physicians in 1539, he soon became rector. Although
his fame as a physician rapidly grew, and many of Europe’s
crowned heads solicited his services, he valued his inde-
pendence too much to become a court physician. In 1543
he accepted a professorship in medicine in Pavia.
Cardano was the most outstanding mathematician of
his time. In 1539 he published two books on arithmetic
embodying his popular lectures, the more important being
Practica arithmetica et mensurandi singularis (“Practice of
Mathematics and Individual Measurements”). His Ars
magna (1545) contained the solution of the cubic equation,
for which he was indebted to the Venetian mathematician
Niccolò Tartaglia, and also the solution of the quartic
equation found by Cardano’s former servant, Lodovico
Ferrari. His Liber de ludo aleae (The Book on Games of Chance)
presents the first systematic computations of probabili-
ties, a century before Blaise Pascal and Pierre de Fermat.
Cardano’s popular fame was based largely on books deal-
ing with scientific and philosophical questions, especially
De subtilitate rerum (“The Subtlety of Things”), a collection
of physical experiments and inventions, interspersed with
anecdotes.
Cardano’s favourite son, having married a disreputable
girl, poisoned her and was executed in 1560. Cardano
never recovered from the blow. From 1562 he was a profes-
sor in Bologna, but in 1570 he was suddenly arrested on the
accusation of heresy. After several months in jail, he was
permitted to abjure privately, but he lost his position and
259
7
Biographies 7
the right to publish books. Before his death he completed
his autobiography, De propria vita (The Book of My Life).
aRThuR cayley
(b. Aug. 16, 1821, Richmond, Surrey, Eng.—d. Jan. 26, 1895,
Cambridge, Cambridgeshire)
The English mathematician Arthur Cayley was the leader
of the British school of pure mathematics that emerged in
the 19th century.
Although Cayley was born in England, his first seven
years were spent in St. Petersburg, Russia, where his
parents lived in a trading community affiliated with the
Muscovy Company. On the family’s permanent return
to England in 1828 he was educated at a small private
school in Blackheath, followed by the three-year course
at King’s College, London. Cayley entered Trinity College,
Cambridge, in 1838 and emerged as the champion student
of 1842, the “Senior Wrangler” of his year. A fellowship
enabled him to stay on at Cambridge, but in 1846 he left
the university to study the law at Lincoln’s Inn in London.
Cayley practised law in London from 1849 until 1863, while
writing more than 300 mathematical papers in his spare
time. In recognition of his mathematical work, he was
elected to the Royal Society in 1852 and presented with
its Royal Medal seven years later. In 1863 he accepted the
Sadleirian professorship in mathematics at Cambridge—
sacrificing his legal career to devote himself full-time
to mathematical research. In that same year he married
Susan Moline, the daughter of a country banker.
Cayley’s manner was diffident but decisive. He was
a capable administrator who quietly and effectively dis-
charged his academic duties. He was an early supporter
of women’s higher education and steered Newnham